Mathematics of Nanosatellites

Abstract

The role of small inexpensive satellites continuously grows in the modern space exploration. Their use can significantly reduce the cost of the mission. However, the control of such satellites is a challenge. One of the major issues here is that such satellites usually do not possess a complex attitude control system, and three-axis stabilization might be unavailable. As a consequence, the thrust vector of the orbit control system cannot be arbitrary oriented in space and rather involved mathematical methods are needed in order to compensate the control system’s simplicity. The most frequently used simple and lightweight passive systems of one-axis stabilization are spin, passive magnetic or aerodynamic stabilization. In this case, one or two orbit control thrusters can be installed along the stabilized axis, so the orientation of the thrust vector at any given moment of time is determined by the orientation of stabilized axis. A very similar situation we face if the satellite uses the radiation pressure from the Sun as a motive force. Recently, a serious progress was achieved in the analysis and solution of the respective control problems. For example, dampers which use magnetic hysteresis rods in order to dissipate the energy of undesired angular motions — occurred during deployment or caused by perturbations — are used in attitude control systems of small satellites since the 1960s (Fischell and Mobley, 1964). The mathematical modeling of such systems is quite a difficult task, since the majority of existent hysteresis models result in differential equations with discontinuous right-hand side. The analysis of dynamics for attitude control systems with magnetic hysteresis dampers and optimization of their parameters have been done in Sarychev et al. (1988) and Guerman et al. (1989), and the results of these studies have been implemented in real missions (Ovchinnikov et al., 2000; Santoni and Zelli, 2009). However, these studies lacked an accurate theoretical basis for application of